Nuprl Lemma : full-arcsin_wf
∀[a:{a:ℝ| a ∈ [r(-1), r1]} ]. (full-arcsin(a) ∈ {x:ℝ| (x ∈ [-(π/2), π/2]) ∧ (rsin(x) = a)} )
Proof
Definitions occuring in Statement :
full-arcsin: full-arcsin(a),
halfpi: π/2,
rsin: rsin(x),
rccint: [l, u],
i-member: r ∈ I,
req: x = y,
rminus: -(x),
int-to-real: r(n),
real: ℝ,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
minus: -n,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
full-arcsin: full-arcsin(a),
int_nzero: ℤ-o,
true: True,
nequal: a ≠ b ∈ T ,
not: ¬A,
implies: P ⇒ Q,
uimplies: b supposing a,
sq_type: SQType(T),
all: ∀x:A. B[x],
guard: {T},
false: False,
prop: ℙ,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
int-rdiv: (a)/k1,
divide: n ÷ m,
int-to-real: r(n),
and: P ∧ Q,
rless: x < y,
sq_exists: ∃x:A [B[x]],
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
nat: ℕ,
le: A ≤ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
req_int_terms: t1 ≡ t2,
rneq: x ≠ y,
rsub: x - y,
radd: a + b,
accelerate: accelerate(k;f),
reg-seq-list-add: reg-seq-list-add(L),
cbv_list_accum: cbv_list_accum(x,a.f[x; a];y;L),
cons: [a / b],
rminus: -(x),
rnexp: x^k1,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
bfalse: ff,
canon-bnd: canon-bnd(x),
absval: |i|,
rdiv: (x/y),
rmul: a * b,
rinv: rinv(x),
mu-ge: mu-ge(f;n),
lt_int: i <z j,
btrue: tt,
imax: imax(a;b),
reg-seq-inv: reg-seq-inv(x),
le_int: i ≤z j,
bnot: ¬bb,
reg-seq-mul: reg-seq-mul(x;y),
fastexp: i^n,
efficient-exp-ext,
genrec: genrec,
subtract: n - m,
nil: [],
it: ⋅,
real: ℝ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rge: x ≥ y,
rgt: x > y,
sq_stable: SqStable(P),
pi: π,
rat_term_to_real: rat_term_to_real(f;t),
rtermVar: rtermVar(var),
rat_term_ind: rat_term_ind,
pi1: fst(t),
rtermMultiply: left "*" right,
rtermConstant: "const",
rtermDivide: num "/" denom,
pi2: snd(t),
stable: Stable{P},
i-member: r ∈ I,
rooint: (l, u),
has-value: (a)↓
Lemmas referenced :
rless-case_wf,
int-rdiv_wf,
subtype_base_sq,
int_subtype_base,
istype-int,
nequal_wf,
int-to-real_wf,
decidable__lt,
full-omega-unsat,
intformnot_wf,
intformless_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-less_than,
rabs_wf,
member_rccint_lemma,
real_wf,
i-member_wf,
rccint_wf,
radd-preserves-rleq,
rsub_wf,
rmul_wf,
radd_wf,
itermSubtract_wf,
itermAdd_wf,
itermMultiply_wf,
itermVar_wf,
rnexp_wf,
istype-le,
square-rleq-1-iff,
rabs-rleq-iff,
rleq_wf,
squash_wf,
true_wf,
rminus-int,
subtype_rel_self,
iff_weakening_equal,
rleq_functionality,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
req_inversion,
rnexp2,
req_weakening,
rdiv_wf,
rless-int,
rless_wf,
rsqrt_wf,
rnexp-rless,
rleq-int-fractions2,
nat_plus_properties,
istype-false,
square-nonneg,
rmul_comm,
image-type_wf,
less_than'_wf,
equal-wf-base,
set_subtype_base,
less_than_wf,
rless_functionality,
int-rdiv-req,
rsqrt-rnexp-2,
req_functionality,
req_transitivity,
rabs-rmul,
rabs-of-nonneg,
rless_functionality_wrt_implies,
rsub_functionality_wrt_rleq,
rleq_weakening_equal,
rleq_weakening_rless,
square-rless-implies,
rsqrt_nonneg,
rless-int-fractions3,
rleq-int-fractions3,
arcsine-bounds,
member_rooint_lemma,
rless_transitivity1,
MachinPi4_wf,
sq_stable__req,
int-rmul_wf,
halfpi_wf,
pi_wf,
assert-rat-term-eq2,
rtermMultiply_wf,
rtermConstant_wf,
rtermDivide_wf,
rtermVar_wf,
rnexp_functionality,
int-rmul_functionality,
int-rmul-req,
rmul_functionality,
rdiv_functionality,
arcsine-nonneg,
member_rcoint_lemma,
partial-arcsin_wf,
arcsine_wf,
i-member_functionality,
rcoint_wf,
radd-preserves-rless,
rminus_wf,
itermMinus_wf,
halfpi-positive,
rocint_wf,
member_rocint_lemma,
rless-implies-rless,
trivial-rsub-rleq,
rleq-implies-rleq,
real_term_value_minus_lemma,
rsub_functionality,
rless_transitivity2,
req_wf,
rsin_wf,
rsin_functionality,
stable_req,
false_wf,
not_wf,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
arcsine-shift,
rsin-arcsine,
not-rless,
rleq_antisymmetry,
rleq_weakening,
rsqrt0,
rsqrt_functionality,
rooint_wf,
arcsine0,
arcsine_functionality,
rsin-halfpi,
rabs-strict-ub,
rless-int-fractions2,
rless_irreflexivity,
rless-int-fractions,
real-has-value,
nat_wf,
le_wf,
absval-non-neg,
absval-minus,
mul-minus,
rneq-int,
intformeq_wf,
int_formula_prop_eq_lemma,
rminus-rminus,
rsin-rminus,
rminus_functionality,
rabs-rless-iff,
rinv_wf2,
iff_weakening_uiff,
efficient-exp-ext
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
setElimination,
thin,
rename,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
dependent_set_memberEquality_alt,
natural_numberEquality,
lambdaFormation_alt,
instantiate,
cumulativity,
intEquality,
independent_isectElimination,
hypothesis,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
voidElimination,
equalityIstype,
baseClosed,
sqequalBase,
universeIsType,
hypothesisEquality,
closedConclusion,
because_Cache,
sqequalRule,
independent_pairFormation,
imageMemberEquality,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
isect_memberEquality_alt,
addEquality,
applyEquality,
inhabitedIsType,
setIsType,
minusEquality,
productElimination,
imageElimination,
universeEquality,
int_eqEquality,
inrFormation_alt,
dependent_set_memberFormation_alt,
productEquality,
baseApply,
productIsType,
unionEquality,
functionEquality,
functionIsType,
unionIsType,
callbyvalueReduce,
promote_hyp
Latex:
\mforall{}[a:\{a:\mBbbR{}| a \mmember{} [r(-1), r1]\} ]. (full-arcsin(a) \mmember{} \{x:\mBbbR{}| (x \mmember{} [-(\mpi{}/2), \mpi{}/2]) \mwedge{} (rsin(x) = a)\} )
Date html generated:
2019_10_31-AM-06_13_48
Last ObjectModification:
2019_05_21-PM-11_09_40
Theory : reals_2
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